Book:Murray R. Spiegel/Theory and Problems of Complex Variables/SI (Metric) Edition/Errata

Errata for 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.)

Example: $\paren {z_3 - \overline {z_3} }^5$

Chapter $1$: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$:

If ... $z_3 = \sqrt 3 - 2 i$, evaluate ...:

$\paren {z_3 - \overline {z_3} }^5$

Ans. $1024 i$

Condition for Points in Complex Plane to form Parallelogram

Chapter $1$: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $65$:

Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices for quadrilateral $ABCD$. Prove that $ABCD$ is a parallelogram if and only if $z_1 - z_2 - z_3 + z_4 = 0$.

Locus represented by $z \paren {\overline z + 2} = 3$

Chapter $1$: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(d)}$:

Describe and graph the locus represented by each of the following:

... $\text (d)$ $z \paren {\overline z + 2} = 3$

Ans. ... $\text (d)$ circle, ...

Polar Form of Complex Number: $3 \, \map \cis {\dfrac {-2 \pi} 3}$

Chapter $1$: Supplementary Problems: Polar Form of Complex Numbers: $84 \ \text{(f)}$:

Graph each of the following and express in rectangular form.

... $\text {(f)} \ \ 3 e^{-2 \pi i / 3}$

Ans. ... $\text {(f)} \ -3 \sqrt 3 / 2 - \paren {3 / 2} i$

Complex Addition: Travel $2$

Chapter $1$: Supplementary Problems: Polar Form of Complex Numbers: $85$:

An airplane travels $150 \, \mathrm {km}$ southeast, $100 \, \mathrm {km}$ due west, $225 \, \mathrm {km}$ $30 \degrees$ north of east, and then $323 \, \mathrm {km}$ northeast. Determine ... how far and in what direction it is from its starting point.

Ans. $\ 375 \, \mathrm {km}$, $23 \degrees$ north of east (approx.)

Complex Division: $\dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {\paren {4 \cis \dfrac {2 \pi} 3}^2}$

Chapter $1$: Supplementary Problems: De Moivre's Theorem: $89 \text {(d)}$:

Evaluate each of the following:

... $\dfrac {\paren {3 e^{\pi i / 6} } \paren {2 e^{- 5 \pi i / 4} } \paren {6 e^{5 \pi i / 3} } } {\paren {4 e^{2 \pi i / 3} }^2}$

Ans. $3 \sqrt 2 / 2 - \paren {3 \sqrt 3 / 2} i$

Quadruple Angle Formula for Sine

Chapter $1$: Supplementary Problems: De Moivre's Theorem: $93 \ \text {(a)}$:

Prove that:

$\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 = 2 \cos 3 \theta + 6 \cos \theta - 4$

5th Roots of $-16 + 16 \sqrt 3 i$

Chapter $1$: Supplementary Problems: Roots of Complex Numbers: $96 \ \text {(c)}$:

Find all the indicated roots and locate them in the complex plane.

... $\text {(c)}$ fifth roots of $-16 + 16 \sqrt 3 i$, ...

Ans. ... $\text {(c)}$ $2 \cis 48 \degrees$, $2 \cis 120 \degrees$, $2 \cis 192 \degrees$, $2 \cis 264 \degrees$, $2 \cis 336 \degrees$ ...

Roots of $z^6 + 1 = \sqrt 3 i$

Chapter $1$: Supplementary Problems: Roots of Complex Numbers: $97 \ \text {(b)}$:

Solve the equations ... $\text{(b)}$ $z^6 + 1 = \sqrt 3 i$

Ans. ... $\set {\sqrt [6] 2 \cis 40 \degrees, \sqrt [6] 2 \cis 1000 \degrees, \sqrt [6] 2 \cis 160 \degrees, \sqrt [6] 2 \cis 220 \degrees, \sqrt [6] 2 \cis 280 \degrees, \sqrt [6] 2 \cis 340 \degrees}$

Cube Roots of $-11 - 2 i$

Chapter $1$: Supplementary Problems: Roots of Complex Numbers: $99$:

Find the cube roots of $-11 - 2 i$.

Ans. ... $1 + 2 i, \dfrac 1 2 - \sqrt 3 + \paren {1 + \dfrac {\sqrt 3} 2} i, -\dfrac 1 2 - \sqrt 3 + \paren {\dfrac {\sqrt 3} 2 - 1} i$

Examples of Set Intersection and Set Union

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $123$:

If $A$, $B$ and $C$ are the point sets defined by $\cmod {z + i} < 3$, $\cmod z < 5$, $\cmod {z + 1} < 4$, represent graphically each of the following:

$\textit {(a)} \quad A \cap B \cap C$, $\quad \textit {(b)} \quad A \cup B \cup C$, $\quad \textit {(c)} \quad A \cap B \cup C$, $\quad \textit {(d)} \quad C \paren {A + B}$, $\quad \textit {(d)} \quad \paren {A \cup B} \cap \paren {B \cup C}$, $\quad \textit {(e)} \quad AB + BC + CA$, $\quad \textit {(f)} \quad A \tilde B + B \tilde C + C \tilde A$

Example of Set Intersection with Union

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $123 \ \text{(c)}$:

If $A$, $B$ and $C$ are the point sets defined by $\cmod {z + i} < 3$, $\cmod z < 5$, $\cmod {z + 1} < 4$, represent graphically ... :

$\textit {(c)} \quad A \cap B \cup C$

Condition for Quartic with Real Coefficients to have Wholly Imaginary Root

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $129$:

$\text{(a)} \quad$ Show that the equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a

pure imaginary root if ${a_3}^2 + {a_1}^2 a_4 = a_1 a_2 a_3$.

Cosine to Power of Odd Integer

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $130 \ \text{(a)}$:

Prove that $\cos^n \phi = \dfrac 1 {2^{n - 1} } \set {\cos n \phi + n \cos \paren {n - 2} \phi + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \phi + \cdots + R_n}$

where $R_n = \begin{cases} \cos \phi & \textit {if $n$ is odd} \\ \dfrac {n!} {\sqbrk {\paren {n / 2}!}^2} & \textit {if $n$ is even.}\end{cases}$